LONG GAPS BETWEEN DEFICIENT NUMBERS

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1 LONG GAPS BETWEEN DEFICIENT NUMBERS PAUL POLLACK Abstract. Let n be a natral nmber. If the sm of the roer divisors of n is less than n, then n is said to be deficient. Let G(x) be the largest ga between consective deficient nmbers belonging to the interval [, x]. In 935, Erdős roved that there are ositive constants c and c 2 with c log log log x G(x) c 2 log log log x for all large x. We rove that G(x)/ log log log x tends to a limit as x, and we describe the limit exlicitly in terms of the distribtion fnction of σ(n)/n. We also rove that long rns of nondeficient nmbers are scarce, in that the roortion of n for which n+,..., are all nondeficient decays trily exonentially in A.. Introdction The ancient Greeks called the ositive integer n deficient, erfect, or abndant, according to whether σ(n) < 2n, σ(n) = 2n, or σ(n) > 2n, resectively. Here σ(n) := d n d is the sal sm of divisors fnction. Denoting these sets D, A, and P, we have D = {, 2, 3, 4, 5, 7, 8, 9, 0,, 3, 4, 5, 6, 7,... }, A = {2, 8, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72,... }, P = {6, 28, 496, 828, , , ,... }. From the analytic standoint, it is natral to ask what roortion of the natral nmbers fall into each of these three classes. From the limited evidence resented above, we might conjectre that the erfect nmbers have density zero, while the abndant and deficient nmbers each make a ositive roortion of the integers. Actally or innocent qestion reqires some care to make recise, as it is not clear a riori that these roortions are well-defined. That this is the case follows from a reslt discovered indeendently by each of Behrend, Chowla [Cho34], and Davenort [Dav33]: Theorem A. For each real nmber, define D() := lim x #{n x : σ(n)/n }. x Then D() exists for all. Moreover, D() is a continos fnction of and is strictly increasing for. Finally, D() = 0 and lim D() = Mathematics Sbject Classification. Primary A25; Secondary N60. Key words and hrases. aliqot arts, deficient nmbers, abndant nmbers, additive arithmetic fnctions. This material is based on work sorted by the National Science Fondation nder agreement No. DMS Any oinions, findings, and conclsions or recommendations exressed in this material are those of the athor and do not necessarily reflect the views of the National Science Fondation.

2 2 PAUL POLLACK The continity of D() imlies immediately that the erfect nmbers make a set of asymtotic density zero. We then dedce that the deficient nmbers make a set of asymtotic density D(2) and that the abndant nmbers make a set of asymtotic density D(2). Deléglise [Del98] has shown that < D(2) < Ths jst nder in 4 natral nmbers are abndant. These reslts adeqately describe the global distribtion of the deficient and abndant nmbers, bt it is reasonable to ask also abot the local distribtion. Often roblems of this tye rove very hard; for examle, is the largest ga between consective rimes in [, x] bonded by O ɛ (x ɛ ) for each ɛ > 0? This, and in fact mch more, is believed to be tre, bt roving this statement seems ot of reach even if one relaces the rimes with their mch denser cosins, the sqarefree integers. For deficient nmbers the sitation is mch better. For x 2, define G(x) as the largest ga n n between consective deficient nmbers n < n x. In 935, Erdős [Erd35] showed the existence of ositive constants c and c 2 with () c log log log x G(x) c 2 log log log x for large x. The reslts of [Erd35], sometimes in slightly weaker form, have been rediscovered mltile times; see, e.g., [Gal86], [GK87], [Sán9], [DKK02]. Other reslts on the local distribtion of σ(n)/n are considered in [Erd58]. Or rimary objective is to fill the ga imlicit in () by roving an asymtotic formla for G(x) as x. Theorem. We have G(x)/ log log log x C (2) C := 2 as x, where D() d. Remark. Calclations carried ot by M. Kobayashi show that C , so that 3.48 C Erdős originally hrased his theorem in terms of long rns of abndant nmbers. We now trn or attention to the qestion of how often sch long rns occr. If x 2 is real and A is a ositive integer, we let N(x, A) denote the nmber of n x for which n +,..., n + A are all nondeficient (erfect or abndant). A theorem of Erdős and Schinzel [ES6, Theorem 3] imlies that for any fixed A, the ratio N(x, A)/x tends to a limit as x. Or second reslt shows that this ratio decays trily exonentially with A. Theorem 2. With C as defined in (2), we have x N(x, A) ex ex ex((c + o())a). Here o() indicates a term that tends to zero as A (niformly in x 2), and the constant imlied by is absolte. Theorem 2 imlies half of Theorem, namely that lim s G(x)/ log log log x /C. The lan for the rest of the aer is as follows: In 2, we rove some lemmas which are sefl in the roofs of both theorems. In 3 we rove Theorem 2, and in 4 we comlete the roof of Theorem by roving that lim inf G(x)/ log log log x /C.

3 LONG GAPS BETWEEN DEFICIENT NUMBERS 3 2. Prearation We begin by recording the following lemma, which is a secial case of [Sch28, Satz I]. Let D (t) denote the density of n with n/σ(n) t, so that D (t) = D(/t) in the notation of Theorem A. Lemma. Sose that f is a fnction of bonded variation on [0, ]. Then as A, ( ) n f f(t) dd (t). A σ(n) n A 0 Given a natral nmber B, define F B as the arithmetic fnction which retrns the B- smooth art of its argment, so that F B (n) := B v(n). (Here we write v (n) for the exonent with the roerty that v(n) n.) Pt H(n) := log σ(n) n and H B (n) := log σ(f B(n)), F B (n) so that H B = H F B. Note that both H and H B are additive fnctions and that H H B ointwise. H B has an imortant near-eriodicity roerty, which we state recisely in the following lemma: Lemma 2. Sose that m and m are natral nmbers with m m (mod M), where M := ( B )B. Then H B (m) H B (m ) 0 as B, niformly in m and m. Proof. Let B be rime. Since v (M) = B and M m m, either v (m) = v (m ) or both v (m) B and v (m ) B. Conseqently, H B (m) H B (m ) = B = log σ(v(m) ) v(m) B v (m) B,v (m ) B B log log σ(v(m ) ) v(m ) ( v(m)+ v(m) v(m ) v(m )+ ) B+, which tends to zero as B (by dominated convergence). Lemma 3. When B, we have M M max{log 2 H B (m), 0} m= where M = ( B )B, as above. 2 D() Proof. By Lemma with f(x) := max{log 2 log x, 0} (intereted so that f(0) = 0), we have M (3) M max{log 2 H(m), 0} m= /2 = log 2 (log 2 + log ) dd () /2 D () d = /2 d, D () d = 2 D() d

4 4 PAUL POLLACK as B, and hence M, tends to infinity. Now notice that 0 M Since (4) log σ(v(m) ) v(m) (max{log 2 H B (m), 0} max{log 2 H(m), 0}) M (H(m) H B (m)) = M the above difference is bonded above by M = M m >B ( = log ) ( v(m) < log + ) <, m >B >B Since B, the reslt follows from (3). m >B ( ) B. log σ(v(m) ) v(m). Combining Lemmas 2 and 3 yields the key estimate for the roofs of both Theorems and 2. Lemma 4. For each A >, t B := (log A) /3. As A, we have A m=n+ niformly for nonnegative integers n. max{log 2 H B (m), 0} 2 D() Proof. Pt M = ( B )B, and notice that by the rime nmber theorem (or a more elementary estimate), M ex(o((log A) 2/3 )). In articlar, M = o(a). We slit the sm in the lemma into a sm over blocks of the form Mk +, Mk + 2,..., M(k + ), exclding O(M) terms at the beginning and end of the original range of smmation. By Lemma 3 and the near-eriodicity established in Lemma 2, we find that M(k+) m=mk+ d, max{log 2 H B (m), 0} = (C + o())m, where C = 2 D() d. Since there are A/M + O() blocks, the total contribtion from all the blocks is (C + o())a. Finally, notice that the contribtion from the O(M) exclded initial and final terms is O(M) = o(a), since each smmand is bonded by log Proof of Theorem 2 By adjsting the imlied constant, we can assme that A is large. We can also assme that A x, since otherwise N(x, A) = 0. To see this last claim, notice that when A > x > n, the interval n +,..., n + A contains not merely a deficient nmber bt in fact a rime nmber. This follows from Bertrand s ostlate, which asserts the existence of a rime in every interval of the form (n, 2n].

5 LONG GAPS BETWEEN DEFICIENT NUMBERS 5 So sose that n is conted by N(x, A) where A is large bt A x. Pt B = (log A) /3. Then by (4) and Lemma 4, (5) m=n+ m >B m=n+ m >B log σ(v(m) ) v(m) = m=n+ m=n+ (H(m) H B (m)) max{log 2 H B (m), 0} (C + o())a, where C is given by (2). To roceed we need a lower bond on N, where (6) N := (n + )(n + 2) (n + A). To obtain or bond, we remove the overla from (5), corresonding to rimes which divide more than one of n +,..., n + A. Clearly any sch satisfies < A, and so the contribtion of sch rimes to the sm in (5) is bonded by B<<A n+ m m 2A >B ( ) 2A B, which is o(a) as A. It follows that we can choose a fnction r(a), deending only on A, with r(a) = o() as A and (C r(a))a. N Let Z = Z(A) be the smallest ositive integer for which (C r(a))a, so that Z = ex(ex((c + o())a)) as A. Z (Here we se the well-known estimate Z = log log Z + O().) We now slit the n conted by N(x, A) into two classes: (i) In the first class, we consider those n for which N, as defined in (6), has at least distinct rime divisors not exceeding 4Z. (ii) In the second class we t all the remaining vales of n. Z 2 log Z If n belongs to the first class, then for some i A, the nmber n + i has at least (2A) Z/ log Z rime divisors not exceeding 4Z. Since n + i 2x, the nmber of ossible n that arise this way is at most A 2x k! 4Z k Z, where k :=, 2A log Z by the mltinomial theorem. A short calclation shows that for large A, this bond is ( ( )) log log log Z ex O Z log Z log log Z x xa ex( Z 4A ) ex ex ex((c + o())a). (To verify this, it is helfl to kee in mind that A log log Z for large A.)

6 6 PAUL POLLACK It remains to show that we have a similar estimate for the n belonging to the second class. Note that for large Z, the first Z/(2 log Z) rimes all belong to the interval [, 2Z/3]. Conseqently, for n in the second class, once A is sfficiently large. Hence, (7) N N >4Z N 4Z 2Z/3 2 3 Z, (C r(a))a 2Z/3 where for the last ineqality we se the minimality of Z. Now (8) Z (π(z ) π(2z/3)) > 4 log Z 2Z/3<<Z N 4 j Z< 4 j+ Z 2Z/3<<Z, when A, and hence Z, is large. It follows from (7) and (8) that there is some j for which 2 j 4 log Z. For this j, the nmber N is divisible by at least 4 log Z rimes from the interval (4j Z, 4 j+ Z], and so one of the nmbers n +,..., n + A is divisible by at least W j := 2j Z 4A log Z of these rimes. By another alication of the mltinomial theorem, we have that the nmber of sch n is at most j A 2x W. W j! 2 j Z 4 j Z< 4 j+ Z Now we sm this exression over j, noting that for large A the inner sm here is bonded by /2. This gives (for large A) an er bond on the nmber n in the second class which is Ax 2 W jw j!. j= For large A, the sm here is dominated by its first term, and we obtain a final bond of as desired. Ax 2 W W! = Ax ex(( + o())w log W ) Ax ex(z/(4a)) X ex ex ex((c + o())a), Remark. This roof ses a method introdced by Erdős [Erd46] and alied by him to estimate D() as. 4. Proof of Theorem Fix ɛ > 0, which we assme to be small. For all large x, we will constrct a ositive integer n x/2 with n +, n + 2,..., n + A all nondeficient, where (9) A := (C + 8ɛ) log log log x.

7 LONG GAPS BETWEEN DEFICIENT NUMBERS 7 If n denotes the first deficient nmber after n, then (for examle, by Bertrand s ostlate) n 2n x, and n n > (n + A) n = A. Ths G(x) > A. Since ɛ > 0 is arbitrary, this imlies the lower bond imlicit in Theorem. (Recall that the er bond follows from Theorem 2.) With B := (log A) /3, we let 0 < <... be the seqence of consective rimes exceeding B. Pt i 0 := 0. If i 0, i,..., i l have been defined, choose i l as small as ossible so that H(P l ) ɛ + max{log 2 H B (l), 0} where P l := j. i l j <i l Sose now that n is chosen so that M := B B n and P l n + l for all l A. Sch a choice is ossible by the Chinese remainder theorem; indeed, the n which satisfy these conditions make a nonzero reside class modlo M A l= P l. For any sch n, and any l A, we have H(n + l) H(P l ) + H B (n + l) H(P l ) + H B (l) ɛ log 2, once x is sfficiently large. (Here we se the near-eriodicity roerty established in Lemma 2 to obtain the last ineqality.) Ths all of n +,..., n + A are nondeficient. To show that sch a choice is ossible with n x/2, it is enogh to show that A M P l 2 x, B< Z l= once x is sfficiently large. Write A l= P l in the form B< Z, where Z is chosen as small as ossible. Then for large A, log σ() A A ( = H(P l ) max{log 2 H B (l), 0} + ɛ + ) B l= l= AC + 2Aɛ + A B = A(C + 3ɛ). (Here the last ineqality ( of the first line follows from or choosing i l minimally at each stage.) Since log σ() = + O ), it follows that 2 A(C + 3ɛ) + O() + log log B A(C + 4ɛ), and hence Conseqently, M Z Z ex ex(a(c + 5ɛ)). l A P l ex(o((log A) 2/3 )) Z ex(o((log A) 2/3 )) ex ex ex(a(c + 6ɛ)) ex ex ex(a(c + 7ɛ)) 2 x

8 8 PAUL POLLACK by or definition (9) of A. Remark. The above argment shows that for each ɛ > 0 and all large x, we have x (0) N(x, A) M l A P x/ ex ex ex(a(c + 7ɛ)) l as x, if A is defined by (9). In fact, or roof shows that if A tends to infinity with x and if A is bonded above by the exression on the right of (9), then the ineqality (0) holds. This can be viewed as a lower-bond conterart of Theorem 2. Acknowledgements I wold like to thank Carl Pomerance for sggesting the roblem of determining an asymtotic formla for G(x). I am also gratefl to Mitso Kobayashi for estimating the vale of C, slying the reference for Lemma, and for giving nice talks that inflenced the writing of the introdction. References [Cho34] S. Chowla, On abndant nmbers, J. Indian Math. Soc. (934), [Dav33] H. Davenort, Über nmeri abndantes, Sitzngsberichte Akad. Berlin (933), [Del98] M. Deléglise, Bonds for the density of abndant integers, Exeriment. Math. 7 (998), no. 2, [DKK02] J.-M. De Koninck and I. Kátai, On the freqency of k-deficient nmbers, Pbl. Math. Debrecen 6 (2002), no. 3-4, [Erd35] P. Erdős, Note on consective abndant nmbers, J. London Math. Soc. 0 (935), [Erd46], Some remarks abot additive and mltilicative fnctions, Bll. Amer. Math. Soc. 52 (946), [Erd58], Some remarks on Eler s ϕ fnction, Acta Arith. 4 (958), 0 9. [ES6] P. Erdős and A. Schinzel, Distribtions of the vales of some arithmetical fnctions, Acta Arith. 6 (960/96), [Gal86] J. Galambos, On a conjectre of Kátai concerning weakly comosite nmbers, Proc. Amer. Math. Soc. 96 (986), no. 2, [GK87] J. Galambos and I. Kátai, The gas in a articlar seqence of integers of ositive density, J. London Math. Soc. (2) 36 (987), no. 3, [Sán9] J. Sándor, On a method of Galambos and Kátai concerning the freqency of deficient nmbers, Pbl. Math. Debrecen 39 (99), no. -2, [Sch28] I. Schoenberg, Über die asymtotische Verteilng reeler Zahlen mod, Math. Z. 28 (928), no., Deartment of Mathematics, University of Illinois, 409 West Green Street, Urbana, Illinois 6802 Crrent address: School of Mathematics, Institte for Advanced Stdy, Einstein Drive, Princeton, New Jersey URL: htt:// address: ollac@illinois.ed